Solving Polynomial and Rational Inequalities
.
Sol 1 Factoring gives
or
. Marking 0,3, and
-2 on a number line, and using that
and that all the
exponents are odd, we get the sign chart shown below:
Therefore the solution is given by
.
Sol 2 Factoring gives or
.
Marking 0,3, and -2 on a number line, and using that
and that the sign changes at 3
and -2 but does not change at 0, we get the sign
chart shown below:
Therefore the solution is given by .
Sol 3 Factoring gives
and that all the exponents are
odd, we get the sign chart shown below:
Since the inequality is not strict, we can include the zeros of the
numerator;
so the solution is given by .
Sol 4 Factoring gives
and that all the exponents are
odd, we get the
sign chart shown below:
Therefore the solution is given by
.
Sol 5 Factoring gives
or
Marking -3,-2,-1,1,2, and 3 on a number line, and using the facts that
and that all the exponents are
odd, we get the following sign chart:
Since the inequality is not strict, we can include the zeros of the
numerator;
so the solution is given by
.
Sol 6 Since for all
,
and therefore
for all
; so multiplying by
gives the equivalent
inequality
.
Factoring yields or
; so marking
on a number
line and using that
when
and all the
exponents are odd,
we get the following sign chart:
Therefore the solution is given by
.
Sol 7 Factoring gives the inequality
Marking off -2,0,1/3,1,3/2, and 4, and using the facts that
and the sign
changes at
3/2,1,-2, and 1/3 and does not change at 0 or at 4, we get the
following sign chart:
Since the inequality is not strict, we can include the zeros of the
numerator;
so the solution is given by
.
Correction: the
solution should include x=0 as well.
Sol 8 Subtracting from both sides
gives
, so
or
.
Marking -4,0,and 4 on a number line, and using that
and that all the exponents are
odd, we get the following sign chart:
Therefore is the solution.
Sol 9 Subtracting from both sides gives
Therefore
Marking 2,4, and -1 on a number line, and using the facts that
and that all the exponents are
odd, we get the
following sign chart:
Therefore is the solution.
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